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Abstract:
Following the direction of 1712.09990 and 1712.09994, this article continues
to excavate more interesting aspects of the 4-particle amplituhedron for a
better understanding of the 4-particle integrand of planar N=4 SYM to all loop
orders, from the perspective of positive geometry. At 3-loop order, we
introduce a much more refined dissection of the amplituhedron to understand its
essential structure and maximally simplify its direct calculation, by fully
utilizing its symmetry as well as the efficient Mondrian way for reorganizing
all contributing pieces. Although significantly improved, this approach
immediately encounters its technical bottleneck at 4-loop. Still, we manage to
alleviate this difficulty by imitating the traditional (generalized) unitarity
cuts, which is to use the so-called positive cuts. Given a basis of dual
conformally invariant (DCI) loop integrals, we can figure out the coefficient
of each DCI topology using its dlog form via positivity conditions. Explicit
examples include all 2+5 non-rung-rule topologies at 4- and 5-loop
respectively. These results remarkably agree with previous knowledge, which
confirms the validity of amplituhedron up to 5-loop and develops a new approach
of determining the coefficient of each distinct DCI loop integral.